On quasi-zero divisor graphs of non-commutative rings

Document Type : Research Paper

Authors

Faculty of Mathematical sciences, Shahrood University of Technology, Shahrood, Iran.

Abstract

Let R be an associative ring with identity. A ring R is called reversible if ab=0, then ba=0 for a,b∈R.
The quasi-zero-divisor graph of R, denoted by Γ∗(R) is an undirected graph with all nonzero zero-divisors of R as vertex set and two distinct vertices x and y are adjacent if and only if there exists 0≠r∈R∖(ann(x)∪ann(y)) such that xry=0 or yrx=0. In this paper, we determine the diameter and girth of Γ∗(R). We show that  the  zero-divisor graph of R denoted by Γ(R), is an induced subgraph of Γ∗(R). Also, we investigate when Γ∗(R) is identical to Γ(R). Moreover, for a reversible ring R, we study the diameter and girth of Γ∗(R[x]) and we investigate when Γ∗(R[x]) is identical to Γ(R[x]).

Keywords

References

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Algebraic Structures and Their Applications
Volume 5, Issue 2
November 2018
Pages 1-13

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